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Infinite circle series

Here’s a fascinating puzzle by Presh Talwalker. Given an isoceles triangle and an infinite series of circles fit within the triangle as depicted below, what is the sum of circumferences of all the circles? Think about it and when you are ready for answer, read on.

Draw a line down the center of the triangle and through all the circles. This line is the sum of diameters of all the circles and is given by the Pythagoras Theorem. The sum of circumferences is therefore:

The puzzle ends here but an even cooler result is implied by the above. If you draw any straight line of length X and fill the entire length with an arbitrary number of circles with arbitrary diameters and centers resting on the line, then as long as there are no gaps between the circles, the sum of circumferences of the circles will always be (Pi)X. The triangle in the puzzle above is just a special case of this wider principle.

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